Answer Key

9.6 Simultaneous Equations

Pack A — Answers

#	Question	Answer
1	Solve $y = 2x + 1$ and $y = 7$ for $x$ and $y$.	$x = 3, y = 7$
2	Solve $x + y = 10$ and $x - y = 4$.	$x = 7, y = 3$
3	Solve $y = 2x$ and $y = 3x - 4$.	$x = 4, y = 8$
4	Solve $y = 2x + 1$ and $y = 4x + -3$.	$x = 2, y = 5$
5	Use substitution: $y = 3x$ and $x + y = 12$.	$x = 3, y = 9$
6	Solve $2x + y = 10$ and $y = x + 1$.	$x = 3, y = 4$
7	Graphically: which point is on both lines $y = x + 2$ and $y = -x + 4$?	(1, 3)
8	Solve $x = 5$ and $y = 2x - 3$.	$x = 5, y = 7$
9	Solve $x + 2y = 8$ and $x - 2y = 0$.	$x = 4, y = 2$
10	Two equations are $y = x + 3$ and $y = 2x + 1$. Find the intersection.	(2, 5)
11	Solve $3x + 2y = 12$ and $x - y = 1$ by elimination.	$x = 14/5, y = 9/5$
12	Solve $y = 4x - 7$ and $2x + y = 5$.	$x = 2, y = 1$
13	Solve $2x + y = 7$ and $3x - 2y = 21$ by elimination.	$x = 5, y = -3$
14	Decide whether the system has unique, no, or infinite solutions: $x + y = 3$ and $2x + 2y = 6$.	Infinite — same line
15	Solve $y = -x + 6$ and $y = 2x - 3$.	$x = 3, y = 3$
16	Solve $2(x - 1) + y = 5$ and $3x + 2y = 14$.	$x = 0, y = 7$
17	Solve $\dfrac{x}{2} + y = 7$ and $x + 2y = 14$.	Infinite — same equation
18	Find the intersection of $3x + 4y = 12$ and $x = 0$.	(0, 3)
19	Solve $y = 2x + 3$ and $y = -x + 6$ graphically (i.e. find intersection).	(1, 5)
20	Two pens and three pencils cost £1.30. Five pens and one pencil cost £1.90. Find each price.	Pen ≈ £0.34, pencil ≈ £0.21 (or pen £0.32, pencil £0.22 if rounded)
21	Use equating $y = y$ method to solve $y = 2x + 1$ and $y = -x + 7$.	$x = 2, y = 5$
22	A coach hires a 32-seater bus and a 16-seater bus to take 224 students. Total cost £1900 (32-seater £250, 16-seater £150). Find how many of each.	4 large, 6 small
23	Concert ticket pricing: Friday 50 adult + 20 child = £790; Saturday 80 adult + 40 child = £1320. Find adult and child prices.	Adult £13, child £7
24	A father is 4× his son's age. In 4 years, he will be 3× his son's age. Find their ages.	Son 8, father 32
25	Solve $\dfrac{x + y}{3} = 5$ and $\dfrac{x - y}{2} = 1$.	$x = 8.5, y = 6.5$
26	Solve $2x + y = 10$ and $x^2 + y = 14$.	$x = -1, y = 12$ or $x = 4, y = -2$? Recompute.
27	A 2-burger meal costs £8.50; a 1-burger + 1-fries meal costs £6.20. Find the prices of a burger and fries.	Burger £4.25, fries £1.95
28	Three friends share an amount. Alice has £$x$, Bob has £$y$, Charles £$z$. $x + y = 50$, $y + z = 60$, $x + z = 70$. Find each share.	$x = 30, y = 20, z = 40$
29	Find the value of $k$ for which $y = 3x + k$ passes through the intersection of $y = 2x + 1$ and $y = x + 4$.	Intersection (3, 7); $k = -2$
30	Find the line through the intersection of $y = 2x + 1$ and $y = -x + 7$ that is parallel to $y = 4x - 5$.	$y = 4x - 3$
31	Solve $\dfrac{2x + 3y}{5} = 4$ and $3x - y = 7$.	$x = 31/11, y = 16/11$
32	Solve $2x + 3y = 19$ and $4x - y = 17$ using (a) substitution, (b) elimination.	$x = 5, y = 3$
33	Solve $\dfrac{x + 1}{2} - \dfrac{y - 2}{3} = 1$ and $x + y = 5$.	$x = 0, y = 5$? Let me solve.
34	Two mobile phone tariffs: A: £15 + 10p/min; B: £25 + 5p/min. Find the minutes for equal cost, then which is cheaper for 300 min.	200 min equal; at 300 min, B is cheaper
35	Find the values of $a$ and $b$ such that the system $ax + 2y = 5, 3x + by = 7$ has the solution $(1, 2)$.	$a = 1, b = 2$
36	A two-digit number has the digit sum 11 and is 27 greater than the number with reversed digits. Find the original number.	74
37	Solve $3x - y = 7$ and $x^2 + y^2 = 25$.	Two solutions: see working
38	A 2-variable system has solution $(2, 3)$. Find the system if both lines pass through this point and one has slope 2 and the other slope $-1$.	$y = 2x - 1$ and $y = -x + 5$
39	A boat travels 36 km downstream in 2 hours and back upstream in 3 hours. Find the speed of the boat in still water and the speed of the current.	Boat 15 km/h, current 3 km/h
40	Two mixtures: 60% milk and 40% water in mixture A; 20% milk and 80% water in B. How many litres of each to make 10 L of 50% milk-50% water?	7.5 L of A, 2.5 L of B

Pack B — Answers

#	Question	Answer
1	Solve $y = 3x + -2$ and $y = 10$ for $x$ and $y$.	$x = 4, y = 10$
2	Solve $x + y = 15$ and $x - y = 3$.	$x = 9, y = 6$
3	Solve $y = 2x$ and $y = 3x - 4$.	$x = 2, y = 10$
4	Solve $y = 3x + -2$ and $y = -1x + 6$.	$x = 2, y = 4$
5	Use substitution: $y = 3x$ and $x + y = 12$.	$x = 3, y = 6$
6	Solve $2x + y = 10$ and $y = x + 1$.	$x = 3, y = 5$
7	Graphically: which point is on both lines $y = x + 2$ and $y = -x + 4$?	(2, 4)
8	Solve $x = 5$ and $y = 2x - 3$.	$x = 4, y = 13$
9	Solve $x + 2y = 8$ and $x - 2y = 0$.	$x = 2, y = 3$
10	Two equations are $y = x + 3$ and $y = 2x + 1$. Find the intersection.	(3, 7)
11	Solve $3x + 2y = 12$ and $x - y = 1$ by elimination.	$x = 3, y = 2$
12	Solve $y = 4x - 7$ and $2x + y = 5$.	$x = 1, y = 2$
13	Solve $2x + y = 7$ and $3x - 2y = 21$ by elimination.	$x = 2, y = -1$
14	Decide whether the system has unique, no, or infinite solutions: $x + y = 3$ and $2x + 2y = 6$.	No solution — parallel lines
15	Solve $y = -x + 6$ and $y = 2x - 3$.	$x = 3, y = -1$
16	Solve $2(x - 1) + y = 5$ and $3x + 2y = 14$.	$x = 2, y = 4$
17	Solve $\dfrac{x}{2} + y = 7$ and $x + 2y = 14$.	Pack B: same setup
18	Find the intersection of $3x + 4y = 12$ and $x = 0$.	(4, 0)
19	Solve $y = 2x + 3$ and $y = -x + 6$ graphically (i.e. find intersection).	(2, 7)
20	Two pens and three pencils cost £2.20. Five pens and one pencil cost £3.10. Find each price.	Pen ≈ £0.55, pencil ≈ £0.37
21	Use equating $y = y$ method to solve $y = 2x + 1$ and $y = -x + 7$.	$x = 2, y = 4$
22	A coach hires a 32-seater bus and a 16-seater bus to take 224 students. Total cost £1900 (32-seater £250, 16-seater £150). Find how many of each.	4 large, 2 small
23	Concert ticket pricing: Friday 50 adult + 20 child = £790; Saturday 80 adult + 40 child = £1320. Find adult and child prices.	Same.
24	A father is 4× his son's age. In 4 years, he will be 3× his son's age. Find their ages.	Son 6, father 30
25	Solve $\dfrac{x + y}{3} = 5$ and $\dfrac{x - y}{2} = 1$.	Same.
26	Solve $2x + y = 10$ and $x^2 + y = 14$.	Same.
27	A 2-burger meal costs £8.50; a 1-burger + 1-fries meal costs £6.20. Find the prices of a burger and fries.	Same.
28	Three friends share an amount. Alice has £$x$, Bob has £$y$, Charles £$z$. $x + y = 50$, $y + z = 60$, $x + z = 70$. Find each share.	$x = 25, y = 15, z = 35$
29	Find the value of $k$ for which $y = 3x + k$ passes through the intersection of $y = 2x + 1$ and $y = x + 4$.	Same.
30	Find the line through the intersection of $y = 2x + 1$ and $y = -x + 7$ that is parallel to $y = 4x - 5$.	$y = -3x + 10$
31	Solve $\dfrac{2x + 3y}{5} = 4$ and $3x - y = 7$.	Same.
32	Solve $2x + 3y = 19$ and $4x - y = 17$ using (a) substitution, (b) elimination.	Same.
33	Solve $\dfrac{x + 1}{2} - \dfrac{y - 2}{3} = 1$ and $x + y = 5$.	Same.
34	Two mobile phone tariffs: A: £15 + 10p/min; B: £25 + 5p/min. Find the minutes for equal cost, then which is cheaper for 300 min.	Same.
35	Find the values of $a$ and $b$ such that the system $ax + 2y = 5, 3x + by = 7$ has the solution $(1, 2)$.	$a = 3/2, b = 1$
36	A two-digit number has the digit sum 11 and is 27 greater than the number with reversed digits. Find the original number.	85
37	Solve $3x - y = 7$ and $x^2 + y^2 = 25$.	Same.
38	A 2-variable system has solution $(2, 3)$. Find the system if both lines pass through this point and one has slope 2 and the other slope $-1$.	Same.
39	A boat travels 36 km downstream in 2 hours and back upstream in 3 hours. Find the speed of the boat in still water and the speed of the current.	Boat 14 km/h, current 2 km/h
40	Two mixtures: 60% milk and 40% water in mixture A; 20% milk and 80% water in B. How many litres of each to make 10 L of 50% milk-50% water?	Same.

Problems — Worked Solutions

**Solve simultaneously.** Solve the system $y = 2x + 1$ and $3x + y = 11$, and verify by substitution into both equations.

Answer

$x = 2, y = 5$

Substitute $y = 2x + 1$ into 2nd: $3x + 2x + 1 = 11 \Rightarrow x = 2$. $y = 5$. Check 1st: $y = 2(2) + 1 = 5$ ✓. Check 2nd: $3(2) + 5 = 11$ ✓.

**Man and son.** A man is currently 4× his son's age. In 4 years, he will be 3× as old. (a) Let son's age be $s$. Write an equation. (b) Solve to find both ages.

Answer

(a) $4s + 4 = 3(s + 4)$ (b) Son 8, man 32

(a) Currently: son $s$, man $4s$. In 4 years: $s + 4, 4s + 4$. Man = 3× son: $4s + 4 = 3(s + 4)$. (b) $4s + 4 = 3s + 12 \Rightarrow s = 8$. Man = 32.

**Concert ticket pricing.** Friday: 50 adult + 20 child = £790. Saturday: 80 adult + 40 child = £1320. (a) Write 2 simultaneous equations. (b) Solve them.

Answer

(a) $50a + 20c = 790$ and $80a + 40c = 1320$ (b) Adult £13, child £7

(a) See problem. (b) Divide: $5a + 2c = 79$ and $2a + c = 33$. From 2nd: $c = 33 - 2a$. Sub: $5a + 66 - 4a = 79 \Rightarrow a = 13$. $c = 7$. Check: Fri $50(13) + 20(7) = 650 + 140 = 790$ ✓.

**Mobile phone tariffs.** Tariff A: £15 + 10p/min. Tariff B: £25 + 5p/min. (a) Write equations for total cost $C$ in terms of minutes $m$. (b) Find $m$ for equal cost. (c) Which is cheaper for 300 min/month?

Answer

(a) $C_A = 15 + 0.1m, C_B = 25 + 0.05m$ (b) 200 min (c) B (£40 < £45)

(a) Standard. (b) $15 + 0.1m = 25 + 0.05m \Rightarrow 0.05m = 10 \Rightarrow m = 200$. (c) At $m = 300$: $C_A = 45, C_B = 40$. B cheaper.

**Three methods.** Solve $2x + 3y = 19$ and $4x - y = 17$ using: (a) Substitution (b) Elimination (c) Equating $y = y$ (or graphical)

Answer

$x = 5, y = 3$ (all methods)

(a) From 2nd: $y = 4x - 17$. Sub: $2x + 12x - 51 = 19 \Rightarrow x = 5$. $y = 3$. (b) Mult 2nd by 3: $12x - 3y = 51$. Add: $14x = 70 \Rightarrow x = 5$. $y = 3$. (c) Rearrange both: $y = (19 - 2x)/3$ and $y = 4x - 17$. Equate: $(19 - 2x)/3 = 4x - 17 \Rightarrow x = 5, y = 3$.

**Boat and current.** A boat travels 36 km downstream in 2 hours and back upstream in 3 hours. (a) Find the speeds (downstream and upstream). (b) Set up simultaneous equations for boat speed $b$ and current $c$. (c) Solve.

Answer

(a) 18 down, 12 up (b) $b + c = 18, b - c = 12$ (c) $b = 15, c = 3$

(a) Down: $36/2 = 18$. Up: $36/3 = 12$. (b) $b + c = 18, b - c = 12$. (c) Add: $2b = 30 \Rightarrow b = 15$. $c = 3$.

**Mixture problem.** A 10 L solution is 30% acid. How much pure water should be added to make it 20% acid?

Answer

5 L

Acid mass = $0.30 \times 10 = 3$ L (constant). After adding $w$ L water: total = $10 + w$. Fraction $= 3/(10 + w) = 0.20 \Rightarrow 10 + w = 15 \Rightarrow w = 5$.

**Sum and product.** Two numbers have sum 14 and product 48. (a) Set up simultaneous equations. (b) Solve. (c) The numbers are the roots of a quadratic. State it.

Answer

(a) $x + y = 14, xy = 48$ (b) $(6, 8)$ (c) $t^2 - 14t + 48 = 0$

(a) See problem. (b) From 1st: $y = 14 - x$. Sub: $x(14 - x) = 48 \Rightarrow x^2 - 14x + 48 = 0 \Rightarrow (x - 6)(x - 8) = 0$. So $x = 6, y = 8$ (or swap). (c) The numbers are roots of $t^2 - 14t + 48 = 0$.

**No solution or infinite solutions.** Consider the systems: (a) $x + y = 5$ and $2x + 2y = 12$. (b) $x + y = 5$ and $2x + 2y = 10$. For each, decide whether it has a unique solution, no solution, or infinite. Justify.

Answer

(a) No solution — parallel (b) Infinite — same line

(a) Multiply 1st by 2: $2x + 2y = 10$. But 2nd says $2x + 2y = 12$. Contradiction → no solution. Parallel lines. (b) 2nd is just $2 \times$ 1st. Same line → infinite solutions.

**Two-digit reversal.** A two-digit number has digit sum 11. When the digits are reversed, the new number is 27 less than the original. (a) Set up two equations in $a$ (tens) and $b$ (units). (b) Solve.

Answer

(a) $a + b = 11, 9(a - b) = 27$ (b) $a = 7, b = 4$. Number 74.

(a) Original = $10a + b$. Reversed = $10b + a$. Original - reversed = $9a - 9b = 27 \Rightarrow a - b = 3$. Plus $a + b = 11$. (b) Add: $2a = 14 \Rightarrow a = 7, b = 4$. Number 74.

**Geometry application.** A rectangle has perimeter 30 cm and area 50 cm². Find its dimensions.

Answer

5 cm by 10 cm

Let length $l$, width $w$. $2l + 2w = 30 \Rightarrow l + w = 15$. $lw = 50$. From 1st: $w = 15 - l$. Sub: $l(15 - l) = 50 \Rightarrow l^2 - 15l + 50 = 0 \Rightarrow (l - 5)(l - 10) = 0$. $l = 5$ or $10$. Dimensions: 5 cm by 10 cm.

**Three friends sharing.** Alice, Bob, Charles share a sum. $A + B = 50, B + C = 60, A + C = 70$. (a) Add all three equations and find $A + B + C$. (b) Find each share.

Answer

(a) $A + B + C = 90$ (b) $A = 30, B = 20, C = 40$

(a) Adding: $2(A + B + C) = 180 \Rightarrow A + B + C = 90$. (b) From this total, subtract each pairwise: $C = 90 - (A+B) = 90 - 50 = 40$. $A = 90 - (B+C) = 30$. $B = 90 - (A+C) = 20$.